INEC said in a statement the court process had delayed its preparations for the Saturday vote for state governors.
In a bid to improve transparency, INEC introduced BVAS for the first time at the national level.
Obi, a former governor, filed to stop INEC from reconfiguring BVAS machines.
The court ruled that the opposition demand could not be granted.
Abuja
Nigeria's electoral agency said on Wednesday it had postponed a weekend governorship election by one week following a court decision over machines used in tallying votes.
The so-called BVAS machines are at the centre of a legal dispute over last month’s contested presidential election which results show Nigeria's ruling APC party candidate won, but the opposition claims it was marred by massive tampering.
With President Muhammadu Buhari stepping down in May after two terms, many Nigerians hoped a clean, transparent vote would usher in a leader capable of tackling growing insecurity and deepening poverty in Africa's most populous nation.
Challenging presidential results
The main opposition Peoples Democratic Party (PDP) and Labour Party say delays in voting in the February 25 presidential ballot and problems in uploading results through the BVAS need to be investigated and are challenging the results of the election.
A court earlier on Wednesday had rejected an opposition demand to halt the reconfiguring of BVAS so their teams could check for forensic evidence of ballot rigging.
The Independent National Electoral Commission (INEC) said in a statement the court process had delayed its preparations for the Saturday vote for state governors.
"While the ruling of the tribunal makes it possible for the commission to commence the preparation of the BVAS for the governorship and state assembly elections, it has come far too late for the reconfiguration," INEC said.
"We thank Nigerians and friends of Nigeria for their understanding as we continue to deal with these difficult issues."
Election delays are not unheard of in Nigeria. In 2019, INEC delayed the presidential election by one week just hours before polling stations opened citing logistical problems.
INEC said the new governorship vote would take place on Saturday, March 18.
Ruling All Progressives Congress (APC) candidate Bola Tinubu, a former Lagos governor, won 8.8 million votes in the race to succeed Buhari, with PDP's Atiku Abubakar at 6.9 million and surprise third challenger Labour Party's Peter Obi at 6.1 million.
In a bid to improve transparency, INEC introduced BVAS for the first time at the national level as well as its IReV, an online database for uploading results from local polling stations.
But some voters and opposition parties said huge delays in voting and failures in the system when uploading tallies allowed for ballot manipulation and disparities from what was counted locally.
Legal tussles
Wednesday's legal ruling was the first volley in what is expected to be a long legal battle over the results of the presidential election.
In 2019, when PDP's Abubakar lost to Buhari, he also challenged the results claiming irregularities. The supreme court dismissed his case months later.
Obi, a former governor, filed to stop INEC from reconfiguring BVAS machines to give his team a chance to inspect them.
The court ruled that the opposition demand could not be granted because INEC needed to carry out the reconfiguration to hold state governor elections.
The court said any data could be safely stored on a backend server.
Data backed up
INEC also said it reassured political parties and candidates that all data from the presidential election would be backed up and available.
Obi, who won over many younger voters with his message that he would usher in change, had been in the Abuja court for the ruling.
The Labour Party candidate said last week he would challenge the outcome of the election in court, claiming he would prove to Nigerians he had won the presidential race.Â
With the third largest number of votes, he managed a significant feat for an outsider in a country where two establishment parties, the APC and PDP, have dominated since the end of military rule in 1999.
PDP candidate Abubakar, who lost his sixth bid for the presidency, denounced Tinubu's victory, describing it as "rape of democracy".
International observers, including from the European Union, also noted logistical problems, disenfranchised voters and a lack of transparency by the INEC.